Another H-super magic decompositions of the lexicographic product of graphs

H Hendy, Kiki A. Sugeng, A.N.M Salman, Nisa Ayunda

Abstract


Let H and G be two simple graphs. The concept of an H-magic decomposition of G arises from the combination between graph decomposition and graph labeling. A decomposition of a graph G into isomorphic copies of a graph H is H-magic if there is a bijection f : V(G) ∪ E(G) → {1, 2, ..., ∣V(G) ∪ E(G)∣} such that the sum of labels of edges and vertices of each copy of H in the decomposition is constant. A lexicographic product of two graphs G1 and G2,  denoted by G1[G2],  is a graph which arises from G1 by replacing each vertex of G1 by a copy of the G2 and each edge of G1 by all edges of the complete bipartite graph Kn, n where n is the order of G2. In this paper we provide a sufficient condition for $\overline{C_{n}}[\overline{K_{m}}]$ in order to have a $P_{t}[\overline{K_{m}}]$-magic decompositions, where n > 3, m > 1,  and t = 3, 4, n − 2.


Keywords


Complement of graph; lexicographic product; H-magic decomposition

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DOI: http://dx.doi.org/10.19184/ijc.2018.2.2.2

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